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Theorem madebdayim 27362
Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)
Assertion
Ref Expression
madebdayim (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴)

Proof of Theorem madebdayim
Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑙 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6925 . . 3 (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ dom M )
2 madef 27331 . . . 4 M :On⟶𝒫 No
32fdmi 6726 . . 3 dom M = On
41, 3eleqtrdi 2844 . 2 (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ On)
5 fveq2 6888 . . . . . 6 (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
6 sseq2 4007 . . . . . 6 (𝑎 = 𝑏 → (( bday 𝑥) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝑏))
75, 6raleqbidv 3343 . . . . 5 (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝑏)( bday 𝑥) ⊆ 𝑏))
8 fveq2 6888 . . . . . . 7 (𝑥 = 𝑦 → ( bday 𝑥) = ( bday 𝑦))
98sseq1d 4012 . . . . . 6 (𝑥 = 𝑦 → (( bday 𝑥) ⊆ 𝑏 ↔ ( bday 𝑦) ⊆ 𝑏))
109cbvralvw 3235 . . . . 5 (∀𝑥 ∈ ( M ‘𝑏)( bday 𝑥) ⊆ 𝑏 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏)
117, 10bitrdi 287 . . . 4 (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏))
12 fveq2 6888 . . . . 5 (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
13 sseq2 4007 . . . . 5 (𝑎 = 𝐴 → (( bday 𝑥) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝐴))
1412, 13raleqbidv 3343 . . . 4 (𝑎 = 𝐴 → (∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝐴)( bday 𝑥) ⊆ 𝐴))
15 elmade2 27343 . . . . . . . 8 (𝑎 ∈ On → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
1615adantr 482 . . . . . . 7 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
17 elpwi 4608 . . . . . . . . . . 11 (𝑙 ∈ 𝒫 ( O ‘𝑎) → 𝑙 ⊆ ( O ‘𝑎))
18 elpwi 4608 . . . . . . . . . . 11 (𝑟 ∈ 𝒫 ( O ‘𝑎) → 𝑟 ⊆ ( O ‘𝑎))
1917, 18anim12i 614 . . . . . . . . . 10 ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)))
20 unss 4183 . . . . . . . . . 10 ((𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)) ↔ (𝑙𝑟) ⊆ ( O ‘𝑎))
2119, 20sylib 217 . . . . . . . . 9 ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙𝑟) ⊆ ( O ‘𝑎))
22 simpr 486 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑙 <<s 𝑟)
23 simplll 774 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑎 ∈ On)
24 dfss3 3969 . . . . . . . . . . . . . . . . 17 ((𝑙𝑟) ⊆ ( O ‘𝑎) ↔ ∀𝑧 ∈ (𝑙𝑟)𝑧 ∈ ( O ‘𝑎))
25 fveq2 6888 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧 → ( bday 𝑦) = ( bday 𝑧))
2625sseq1d 4012 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 → (( bday 𝑦) ⊆ 𝑏 ↔ ( bday 𝑧) ⊆ 𝑏))
2726rspccv 3609 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → (𝑧 ∈ ( M ‘𝑏) → ( bday 𝑧) ⊆ 𝑏))
2827ralimi 3084 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → ∀𝑏𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday 𝑧) ⊆ 𝑏))
29 rexim 3088 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑏𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday 𝑧) ⊆ 𝑏) → (∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → (∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3130adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
32 elold 27344 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏)))
3332adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏𝑎 𝑧 ∈ ( M ‘𝑏)))
34 bdayelon 27258 . . . . . . . . . . . . . . . . . . . . 21 ( bday 𝑧) ∈ On
35 onelssex 6409 . . . . . . . . . . . . . . . . . . . . 21 ((( bday 𝑧) ∈ On ∧ 𝑎 ∈ On) → (( bday 𝑧) ∈ 𝑎 ↔ ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3634, 35mpan 689 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ On → (( bday 𝑧) ∈ 𝑎 ↔ ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3736adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (( bday 𝑧) ∈ 𝑎 ↔ ∃𝑏𝑎 ( bday 𝑧) ⊆ 𝑏))
3831, 33, 373imtr4d 294 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) → ( bday 𝑧) ∈ 𝑎))
3938ralimdv 3170 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (∀𝑧 ∈ (𝑙𝑟)𝑧 ∈ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
4024, 39biimtrid 241 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ((𝑙𝑟) ⊆ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
4140imp 408 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎)
4241adantr 482 . . . . . . . . . . . . . 14 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎)
43 bdayfun 27254 . . . . . . . . . . . . . . . . 17 Fun bday
44 oldssno 27336 . . . . . . . . . . . . . . . . . . 19 ( O ‘𝑎) ⊆ No
45 sstr 3989 . . . . . . . . . . . . . . . . . . 19 (((𝑙𝑟) ⊆ ( O ‘𝑎) ∧ ( O ‘𝑎) ⊆ No ) → (𝑙𝑟) ⊆ No )
4644, 45mpan2 690 . . . . . . . . . . . . . . . . . 18 ((𝑙𝑟) ⊆ ( O ‘𝑎) → (𝑙𝑟) ⊆ No )
47 bdaydm 27256 . . . . . . . . . . . . . . . . . 18 dom bday = No
4846, 47sseqtrrdi 4032 . . . . . . . . . . . . . . . . 17 ((𝑙𝑟) ⊆ ( O ‘𝑎) → (𝑙𝑟) ⊆ dom bday )
49 funimass4 6953 . . . . . . . . . . . . . . . . 17 ((Fun bday ∧ (𝑙𝑟) ⊆ dom bday ) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5043, 48, 49sylancr 588 . . . . . . . . . . . . . . . 16 ((𝑙𝑟) ⊆ ( O ‘𝑎) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5150adantl 483 . . . . . . . . . . . . . . 15 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5251adantr 482 . . . . . . . . . . . . . 14 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → (( bday “ (𝑙𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙𝑟)( bday 𝑧) ∈ 𝑎))
5342, 52mpbird 257 . . . . . . . . . . . . 13 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday “ (𝑙𝑟)) ⊆ 𝑎)
54 scutbdaybnd 27296 . . . . . . . . . . . . 13 ((𝑙 <<s 𝑟𝑎 ∈ On ∧ ( bday “ (𝑙𝑟)) ⊆ 𝑎) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
5522, 23, 53, 54syl3anc 1372 . . . . . . . . . . . 12 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
56 fveq2 6888 . . . . . . . . . . . . 13 ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘(𝑙 |s 𝑟)) = ( bday 𝑥))
5756sseq1d 4012 . . . . . . . . . . . 12 ((𝑙 |s 𝑟) = 𝑥 → (( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎 ↔ ( bday 𝑥) ⊆ 𝑎))
5855, 57syl5ibcom 244 . . . . . . . . . . 11 ((((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((𝑙 |s 𝑟) = 𝑥 → ( bday 𝑥) ⊆ 𝑎))
5958expimpd 455 . . . . . . . . . 10 (((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) ∧ (𝑙𝑟) ⊆ ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎))
6059ex 414 . . . . . . . . 9 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ((𝑙𝑟) ⊆ ( O ‘𝑎) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎)))
6121, 60syl5 34 . . . . . . . 8 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎)))
6261rexlimdvv 3211 . . . . . . 7 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday 𝑥) ⊆ 𝑎))
6316, 62sylbid 239 . . . . . 6 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) → ( bday 𝑥) ⊆ 𝑎))
6463ralrimiv 3146 . . . . 5 ((𝑎 ∈ On ∧ ∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏) → ∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎)
6564ex 414 . . . 4 (𝑎 ∈ On → (∀𝑏𝑎𝑦 ∈ ( M ‘𝑏)( bday 𝑦) ⊆ 𝑏 → ∀𝑥 ∈ ( M ‘𝑎)( bday 𝑥) ⊆ 𝑎))
6611, 14, 65tfis3 7842 . . 3 (𝐴 ∈ On → ∀𝑥 ∈ ( M ‘𝐴)( bday 𝑥) ⊆ 𝐴)
67 fveq2 6888 . . . . 5 (𝑥 = 𝑋 → ( bday 𝑥) = ( bday 𝑋))
6867sseq1d 4012 . . . 4 (𝑥 = 𝑋 → (( bday 𝑥) ⊆ 𝐴 ↔ ( bday 𝑋) ⊆ 𝐴))
6968rspccv 3609 . . 3 (∀𝑥 ∈ ( M ‘𝐴)( bday 𝑥) ⊆ 𝐴 → (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴))
7066, 69syl 17 . 2 (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴))
714, 70mpcom 38 1 (𝑋 ∈ ( M ‘𝐴) → ( bday 𝑋) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cun 3945  wss 3947  𝒫 cpw 4601   class class class wbr 5147  dom cdm 5675  cima 5678  Oncon0 6361  Fun wfun 6534  cfv 6540  (class class class)co 7404   No csur 27123   bday cbday 27125   <<s csslt 27262   |s cscut 27264   M cmade 27317   O cold 27318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-1o 8461  df-2o 8462  df-no 27126  df-slt 27127  df-bday 27128  df-sslt 27263  df-scut 27265  df-made 27322  df-old 27323
This theorem is referenced by:  oldbdayim  27363  madebday  27374
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